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Driving point impedance and voltage ratio, bridged RL network

Short answer

Driving-point impedance Z11 = 1 Ω, a constant resistance. Voltage ratio transfer function G21(s) = V2/V1 = (s+1)/(s^2+3s+1).

For the two port network shown below, find the driving point impedance of port one and the voltage ratio transfer function. [Figure: port 1 series arm: a 1 H inductor in series with the parallel combination of (1 H inductor in series ... a 1 H inductor in parallel with a 1 Ω resistor), the whole arm in parallel with a 1 Ω resistor. Shunt branch at port 2: a 1 Ω resistor in parallel with [a 2 F capacitor in series with the parallel combination of a 1/4 Ω resistor and a 2 F capacitor]. Ports: V1(S), I1(S) and V2(S), I2(S).]

Driving point impedance of port one Z11 = V1/I1 and voltage ratio G21 = V2/V1, both with port 2 open (I2 = 0, assumption).

Reading the figure. Series path from port 1 to the output node: 1 H in series with the parallel combination (1 H ∥ 1 Ω); this whole path is in parallel with a 1 Ω resistor. Shunt arm at port 2: 1 Ω in parallel with [2 F in series with (1/4 Ω ∥ 2 F)].

Series arm

(1 H ∥ 1 Ω) = s/(s + 1)
Zm = s + s/(s + 1) = s(s + 2)/(s + 1)
Zser = 1 ∥ Zm = Zm/(1 + Zm) = s(s + 2)/(s² + 3s + 1)

Shunt arm

1/4 Ω ∥ 2F = (1/4)(1/2s) / (1/4 + 1/2s) = 1/[2(s + 2)]
Zc = 1/(2s) + 1/[2(s + 2)] = (s + 1)/[s(s + 2)]
Zsh = 1 ∥ Zc = Zc/(1 + Zc) = (s + 1)/(s² + 3s + 1)

Driving point impedance (I2 = 0)

Z11 = Zser + Zsh = [s² + 2s + s + 1]/(s² + 3s + 1)

Z11(s) = 1 Ω (a pure, frequency-independent resistance).

Voltage ratio

The output is the voltage across Zsh (open circuit), a divider:

G21 = Zsh/(Zser + Zsh) = Zsh/Z11

G21(s) = (s + 1) / (s² + 3s + 1)

Poles: s = (−3 ± √5)/2 = −0.382, −2.618 (both real, left half-plane); zero at s = −1; G21(0) = 1.

✓ Answer checked by Claude Opus 5.5 (high effort)

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